ISRAEL JOURNAL OF MATHEMATICS, Vol. 67, No. 2, 1989
A COMBINATORIAL CHERN-WEIL THEOREM
FOR 2-PLANE BUNDLES
WITH EVEN EULER CHARACTERISTIC
BY
ETHAN D. BLOCH*
Department of Mathematics, Bard College, Annandale-on-Hudson, NY 12504, USA
ABSTRACT
A combinatorial Chern-Weil theorem for arbitrary oriented 2-plane bundles
with even Euler class over surfacesis proved. Alongthe way a simple method
is developedto use exterior angles to calculate the curvature at the vertices of
a large class of non-convex, non-immersed surfacesin R3.
1.
Preliminaries
Various methods have been developed for calculating combinatorial characteristic classes for tangent and normal bundles of manifolds (see [HT], [Ba2],
[BM], [GT] for a few examples). Recently, partly due to the interest of
physicists in what they refer to as Lattice Gauge Theory, there have been
efforts to work combinatorially directly with arbitrary bundles, as in [PS]. (It is
true that all bundles can be dealt with via tangent and normal bundles, but it
seems desirable to deal with arbitrary bundles directly.) It would be nice to
have an approach based on curvature (as opposed to, say, obstruction theory),
thus allowing for a combinatorial Chern-Weil type theory for arbitrary
bundles. In this paper we develop such a theory for oriented 2-plane bundles
with even Euler characteristic over surfaces. Our procedure has two stages:
first, one passes from smooth vector bundles to a combinatorial analog of
vector bundles; second, one computes Euler classes in a purely combinatorial
manner from the combinatorial bundles. To emphasize that this last step is
indeed combinatorial, we will do it first. (This whole procedure is analogous to
t Partially supported by NSF Contract DMS-8503388.
Received November 20, 1987 and in final revised form April 9, 1989
193
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ETHAN D. BLOCH
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the two stages used to compute the simplicial homology of smooth manifolds:
first one triangulates the smooth manifold, and then one computes the
simplicial homology of the triangulation - - the latter step being the purely
combinatorial one.) It is hoped that the present methods will generalize to
higher dimensions and codimensions.
Partly because of a needed result, and partly for motivation, we will discuss
the following topic in polyhedral geometry before proceeding to characteristic
classes. There are (at least) three ways to measure the curvature at the vertices
of a polyhedral surface in 113. The simplest way is to use the angle defect
d~ -- 2x - X ~ a, where the a are the angles of the triangles containing vertex
v. The properties of d~ are well known (see [G 1] for example). (Note that the
angle defect does have a smooth analog in terms of the perimeters of geodesic
disks.) The second method of finding the curvature of a polyhedral surface is
the Morse-theoretic approach of [Ba 1] and [Ba3], which works very nicely (and
which also has a smooth analog). The disadvantage of both these methods is
that they do not correspond to the usual way one thinks of curvature for
smooth surfaces (e.g. Gauss' original approach); also, they are not useful for
our treatment of characteristic classes. The third method for finding polyhedral curvature is by exterior angles, which is analogous to Gauss' approach to
smooth surfaces. Exterior angles have been widely studied by combinatorialists for convex polyhedra in all dimensions (see [G 1] and [G2] for references).
In w 6 and 7 of this paper we give a completely elementary treatment of
exterior angles that works for a reasonable class of non-convex, non-immersed
surfaces in R 3, and which seems to shed some light on the geometry of such
surfaces. Hopefully this approach will generalize to higher dimensions. For
those only interested in polyhedral geometry, the relevant sections (w 6 and 7)
are independent of the rest of the paper (though the reverse is not true).
The outline of the paper is as follows. In w we state some definitions and
results concerning polyhedral geometry of surfaces. These results are proved in
w and 7. In w we discuss 2-plane bundles; in w we discuss combinatorial
analogs of smooth vector bundles; in w our combinatorial Chern-Weil
theorem is proved.
2.
Extrinsic curvature for polyhedral surfaces
SOME NOTATION. Let S + be an open hemisphere in S 2, given the usual
orientation. If x, y, z E S +, let (x, y) denote the geodesic arc with x and y its
Vol. 67, 1989
A COMBINATORIAL CHERN-WEIL THEOREM
195
endpoints, and let (x, y, z) denote the geodesic triangle with x, y and z its
vertices.
DEFINITION. Given three points x, y, z E S +, let the signed angle a(x, y, z)
be the angle (in [ - rt, to]) from ix, y) t o / z , y), positive or negative depending
upon whether ( x , z , y ) is positively or negatively oriented. Also, let
Area/x, z, y) be the signed area of the triangle ix, z, y), positive or negative
depending upon whether (x, z, y) is positively or negatively oriented.
DEFINITION. Let C, = (al,. 9 9 a,) be a triangulation of S 1, with a given
orientation, and let f : C, ~ S + be a simplexwise geodesic (abbreviated SG)
map (i.e f is a unit speed geodesic on each interval [ak, ak+ 1]). Furthermore,
a s s u m e f is injective on each interval. Let x E S + be any point. Define the area
off(C,) with respect to x to be
A(f, x) = ~ Areaix, f(ak),f(ak+l)).
k=l
THEOREM 2.1. Let S + be an open hemisphere in 8 2, and let f: C, --- S + be
an SG map as above. Then A ( f, x) is independent of the choice of x E S +.
This theorem is proved in w We now apply the theorem to polyhedral
surfaces. To allow for more interesting geometric situations than is possible
with embedded complexes, we make the following
DEFINITION. Let /~ be an abstract n-dimensional simplicial complex,
and let f : (vertices of / ~ } - ' R k be a map such that if {Vo. . . . . v,} are
the vertices of an n-simplex in/~, then { f ( v 0 ) , . . . , f(v,)} are affinely independent in R k. Let K denote the collection of simplices in R k with vertices
{ f(vo),..., f(vp)}, for every simplex {Vo. . . . . %} in/~; we call K a simplexwise
embedded complex in R k.
From now on we will restrict our attention to closed, oriented, simplexwise
embedded surfaces in R 3. In order to have a meaningful notion of extrinsic
curvature, we need some local restrictions somewhat analogous to being
smooth, and quite similar to the transverse fields originally considered by [C],
and more recently by [L].
DEFINITION. Let K C R 3 be an oriented simplexwise embedded surface.
For any simplex q ~ K , q* will denote its barycenter. K ~ will denote the
set of i-simplices of K. For each 2-simplex cr ~ K , define G(tT*)~S 2 to be the
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ETHAN D. BLOCH
Isr. J. Math.
unit normal to tr, corresponding to the orientation of tr inherited from the
orientation of K. K is called star normal at vertex v ~ K if the set
{G(a*) [ a ~ K ~ n star(v, k)} is contained in an open hemisphere in $2; K is
called star normal if it is star normal at all its vertices. If K is star normal, we
then choose, for each vertex v ~ K , some point G(v) in the geodesic convex hull
of the set {G(a*) [ tr ~ K ~2~A star(v, K)}. For any such set of choices, we call the
map G" {a* I o ' ~ K (2)} to K(~ S 2 a combinatorial Gauss map on K.
REMARKS. (1) There are some useful equivalent definitions of star normality. For example, star normality at a vertex v E K i s equivalent to the existence
of a vector G(v)~S 2 so that for each 2-simplex a~star(v, K), orthogonal
projection from G(a*) to the line containing G(v) is an orientation preserving
injection. This definition is seen to be equivalent to the existence of an
oriented plane P~ containing v so that for each 2-simplex a Estar(v, K),
orthogonal projection from a to P~ is an orientation preserving injection.
Finally, for any vector x ~ S ~, let Sx c S 2 denote the open hemisphere centered
at x. Then star normality at v is equivalent to the condition that
O {Sa(~.)I a ~ K ~) N star(v, K)} § ~ .
(2) If G(v) is in the geodesic convex hull of the set
{G(a*) I a E K (2) r star(v, K)},
then G(a*)~Sa(~) for all 2-simplices cr in star(v, K).
(3) Not all polyhedral surfaces are star normal. For example, approximate
the seam of a baseball with a polyhedral curve, and take the cone on this curve
from the center of the ball. However, it is seen that a fine enough
C~176
of a smooth surface in R 3 will be star normal (simply project
onto the tangent plane).
DEFXNmON. Let K be as above, let G be a choice of Gauss map, and let
v ~ K be a vertex. Let P~ be as in Remark (1) above. Orthogonally project
star(v, K) onto P~. Let wrap(v, K) be defined to be the winding number of the
projection of link(v, K) about the origin in P~. (Equivalently, wrap(v, K) is the
number of preimages in star(v, K) under the projection of any point sufficiently near the origin in P, .) Let { r / i , . . . , r/p} be the 2-simplices of star(v, K) in
order corresponding to the orientation of K. Choose an open hemisphere
S + E S 2 containing the set
{G(v)} tO {G(a*) I a ~ K (z) f) star(v, K)},
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A COMBINATORIAL CHERN-WEIL THEOREM
197
which exists by the definition of star normality. Let Cu = (al . . . . , ap) be a
triangulation o f S 1, and let f : Cp ---,S~t~)be the SG map given byf(a~) = G(r/*).
Finally, let A (v, K) be defined by A (v, K) = A ( f , G(v)). We define the extrinsic
curvature e, at v to be
e~ = A(v, K) - 2rt[wrap(v, K) - 1].
LEMMA 2.2. Let K be a closed, oriented, star normal, simplexwise embedded surface in R 3, and let G be a combinatorial Gauss map on K. For all
vertices v ~ K , the quantities A(v, K), wrap(v, K) and e~ are independent o f the
choice o f combinatorial Gauss map.
PROOF. It suffices to show that each of A(v, K) and wrap(v, K) is independent of the choice of combinatorial Gauss map, for all vertices v ~ K . For
A (v, K) this follows immediately from Theorem 2.1. For wrap(v, K), the point
is to observe that the set of all possible choices of G(v) is a geodesically convex
subset of an open hemisphere of S 2 (by definition). Call this set Q. If x, y ~ Q
are two choices for G(v), connect them with a path in Q. It is now easy to use
this path in Q to construct a homotopy of the appropriate projections of
link(v, K), and the lemma follows.
[]
The following result, proved in w shows that e~ equals the angle defect
referred to in w1.
THEOREM 2.3. Let K be a closed, oriented, star normal, simplexwise
embedded surface in R 3. For each vertex v ~ K, e, = 2rt - Z~a a, where the a are
the angles o f the tringles containing vertex v.
REMARKS. (1) The need for the wrapping number in the definition of ev is
due to the existence of simplexwise embedded surfaces which "do not look
like" C~176
of smoothly immersed surfaces in R 3. It is seen that a
C~-triangulation K o f a smooth surface in R 3 has wrap(v, K) = 1 at all vertices.
(2) Theorem 2.3 shows that e~ is invariant under simplexwise linear local
isometries, since that result is evident for the angle defect. (A simplexwise
linear map (SL for short) is a map from a simplicial complex into R ~ which is
affine linear on each closed simplex. An SL map is a local isometry if it
preserves the lengths of 1-simplices.) The following example shows that
neither wrap(v, K) nor A (v, K) alone are invariant under SL isometries.
EXAMPLE. Let B4 (respectively Ba) be a square (resp. octagon) in the
x - y plane, with sides of unit length and center at the origin. Let 2 =
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ETHAN D. BLOCH
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(~/14 - 2,r
and let K be the suspension of B8 from points (0, 0, 2) and
(0, 0 , - 2). Define an SL map f : K----R3 by wrapping B8 twice around B4
(taking vertices to vertices), and havingri(0, 0, + 2)) = (0, 0, + 2). The collection of simplices f(K) is a simplexwise embedded complex. The choice of 2
makes f an SL local isometry K ~ r i K ) . However, it is easy to see that
wrap((0, 0, 2), K) = 1, whereas wrap(r/0, 0, 2), f(K)) = 2. Also, by considering the angle defect d~, it is seen that
A(f(O, O, 2), f(K)) = A((0, 0, 2), K) + 2n.
Are A (v, K) and wrap(v, K) invariant under SL local isometries
that are injective, or locally injective?
QUESTION.
3.
Remarks on 2-plane bundles
From this point on, assume all manifolds are closed, smooth and oriented,
all maps between manifolds are smooth and orientation preserving, and all
bundles are smooth and oriented. Some notation: Gm(Rk) is the Grassmannian
of m-planes in R k, Gm(Rk) is the Grassmannian of oriented m-planes in Rk,
and ?m(Rk) (resp. ~m(Rk)) is the canonical bundle over Gm(Rk) (resp. Gm(Rk)).
Gl(m, R) is the general linear group of real m • m matrices; Gl+(m, R) is the
subset of Gl(m, R) of matrices of positive determinant. If ~ is an m-plane
bundle, let B (~) denote the base space of ~, and let Z(~) E H m(B(~), R) denote
the Euler class of ~.
Let ~ be a smooth (oriented) m-plane bundle. It is a standard result that ~ can
be classified by maps g: B(~) ~ Gm(Rk) for sufficiently large k; in other words,
~ g*(~m(Rk)) for such maps g. In our present treatment, classifying maps are
used to give bundles geometric structure, necessary for a Chern-Weil type
theorem.
It follows from the definition that for an m-plane bundle ~ to be classified in
Gm(R"+~), ~ must be isomorphic to g*(~,m(Rm+l))=g*(TSm) for some
m a p g : B ( ~ ) ~ G m ( R m + l ) ~ S m. Therefore, Wk(~)=0 for k §
or m, and
PK(~) = 0 for k § 0 or m/4 (the latter only if m/4 is an integer). Whereas this is
clearly a very restrictive condition in general, the following lemma gives one
situation where such classifications are possible. Recall that Gz(R 3) -~ S ~.
LEMMA 3.1. Let ~ be a 2-plane bundle over a surface. The following are
equivalent.
(1) Z(~) is even.
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A COMBINATORIAL CHERN-WEIL THEOREM
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(2) ~ is isomorphic to the pullback o f the tangent bundle o f S 2 via some map
B( ~) ~ S 2, and any two such maps are (smoothly) homotopic.
PROOF. (1) =* (2). Smooth, oriented 2-plane bundles over smooth, oriented
surfaces are completely classified by their Euler classes (see [DW]; they call the
Euler class the "integral WE"). By hypothesis X(~) = 2p [B(~)] for some integer
p, where [B(~)] is the fundamental cohomology class in H2(B(~), R). For any
smooth, oriented surface there exists a smooth map g : B(~)-~ S 2 of degree p.
Then
x(g*(TS2)) = g*(z(TS2)) = g*(2[$2]) = 2p[B(~)],
and hence ~ ~-. g*(TS2).
Suppose g, h : B(~)--* S z are maps such that g*(TS 2) ~ h*(TS2). Then
2(deg(g))[B(~)] = 2g*([$1]) = g*(2[SZ]) = g*(z(TS2)) = x(g*(TSE))
= z(h*(TS2)) . . . . .
2(deg(h))[B(~)].
Therefore deg(g)--deg(h), and hence g is homotopic to h by the HopfWhitney Theorem [Wh, p. 244], which states that [B(~), S ~] ~ H2(B(~), R),
where [B(~), S 2] denotes the homotopy classes of maps B(~) --- $2; the isomorphism is given by g w-~ (deg(g))[B(~)]. By smoothing theory, g is in fact
smoothly homotopic to h.
(2)=~ (1). This is straightforward.
[]
REMARK. The homotopy equivalence of maps in the lemma is not entirely
trivial, since in general the proof of the homotopy classification of bundles
involves going into higher dimensional Grassmannians (see [H, p. 32] for
example).
A definition we will need later is
DEFINITION. Let ~ be a smooth m-plane bundle, and let g : B(~) ~ Gm(Rk)
be a classifying map for ~. I f N i s a manifold, and h : N - , B ( ~ ) is a map, then
the bundle h*(~) over N is given the induced classifying map g o h : N --, G,, (R k).
(Since (g o h)* = h* o g*, the map g o h : N ~ G,,(R k) really is a classifying map
for h*(~).)
4.
Curvature and Euler class for 2-dimensional lattice gauge fields
In order to calculate characteristic classes combinatorially, we will replace a
smooth bundle by a combinatorial object (analogous to the process of triangu-
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ETHAN D. BLOCH
Isr. J. Math.
lating a smooth manifold). Such an object is called a lattice gaugefield (LGF
for short), as defined below. (The name "lattice gauge field" was coined by
physicists.) A related combinatorial object we use is a lattice classifying map
(LCM for short), also defined below. We will then define curvature, and an
Euler class, in this combinatorial situation. All homology and cohomology has
coefficients in R, and is simplicial, cellular or singular depending on the
situation in the obvious way.
DEFINITION. Let K be a simplicial complex. K' will denote the first
barycentric subdivision of K; for any simplex r/EK, r/* will denote its
barycenter. A lattice classifying map O/K into Gm(Rk) (called an (m, k)-LCM
for short) is a map O:(K')t~
k) for some k > m. An oriented (m, k)LCM O/K is a map O : K ' ) t ~ Gm(Rk), with the added condition that for
every 1-simplex t / = (z*, a * ) ~ K ' , orthogonal projection from O(r*) to O(a*)
is an orientation preserving injection. A Gl(m, R)- valued lattice gaugefield on
K' (called an m-LGF for short) ~/K is a map f~ : (K') tl~---,Gl(m, R), and an
oriented m-LGF is a map f~ : (K')tl~-- Gl+(m, R). Given an oriented (m, k)LCM O/K, one obtains an induced oriented m- LGF f~/K as follows. For each
vertex a* ~ K ' (a a simplex in K), choose an arbitrary orientation preserving
orthogonal map r~. : O(a*) --- R m. Then, for every 1-simplex r / = (z*, a*) EK'
(where we let r have higher dimension than a), let ~(t/) : R m ~ R m be defined
by f~(r/)= r,. o l-I,,, o (r,.) -1, where YI,~ is orthogonal projection in R k from
O(z*) to O(a*). (Another approach to constructing induced m-LGF's would be
to use Riemannian parallel transport in Gm(Rm+ 1).)
Two ways of obtaining m-LGF's are as follows. (1) Let K be an orientable
simplexwise embedded m-manifold in R k. An (m, k)-LCM O/K is called a
tangent (m, k)-LCM for K if it is an oriented (m, k)-LCM such that O(r*) is
parallel to z for all simplices z E K (and has the same orientation as r if z is an
m-simplex). Not every K has a tangent (m, k)-LCM (orientability is the issue
here), but it is evident that suitable restrictions on K, similar in nature to the
definition of star normality in w will insure the existence of a tangent (2,3)LCM for a surface that is simplexwise embedded in R 3. If K has a tangent
(m, k)-LCM, one then constructs an induced tangent m-LGF from the (m, k)LCM as before. We will denote tangent (m, k)-LCM's and m-LGF's by OT/K
and ~ T / K respectively. It should be noted that i f K is simplexwise embedded
but not locally embedded (i.e. embedded on the star of every vertex), then the
tangent (m, k)-LCM and m-LGF of K will not necessarily correspond to the
tangent bundle of K.
VO1.67, 1989
A COMBINATORIALCHERN-WEIL THEOREM
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(2) Let ~ be a smooth m-plane bundle, and suppose a classifying map
g: B ( ~ ) ~ G,,(R k) for ~ is chosen. If t : K-,B(~) is a C~-triangulation, then
the assignment O ( a * ) = g(t(a*)) for each simplex a ~ K is an (m, k)-LCM,
called O(~, g)/K. If t is a fine enough triangulation, then O(~, g)/K will be
oriented, and will thus give rise to an oriented m-LGF, called D(~, g)/K. We
will refer to O(~, g)/K and ~(~, g)/K as being induced by g and t.
In order to make use of the ideas of w we will restrict our attention from
now on to oriented (2,3)-LCM's. K will always denote a compact, oriented,
triangulated surface.
DEFINITION. Let K be as above, and let O/K be an oriented (2,3)-LCM. If
N E G2(R 3) is a 2-plane, let N *~be the (oriented) normal vector to N. We say O
is simplex-normal if for each 2-simplex a = (u, v, w) ~ K, there is an open
hemisphere H~ c S 2 such that
{O(t/*) * [ t/Estar(u, K) U star(v, K) U star(w, K)} c H~.
Now assume O/K is simplex-normal. Let vEK be a vertex. Suppose the
vertices of link(v, K') are {x~, Yt. . . . , xp, yp} in order corresponding to the
orientation of K, where the Xk are the barycenters of the 2-simplices, and the Yk
are the barycenters of 1-simplices. Choose some 2-simplex a Estar(v, K);
define an SG map f : link(v,K')~H~ by setting f(z)= O(z) * for each z E
{x~, y~. . . . , xp, yp }. We define the curvature of O/K at v to be Tv = A (f, O(v)#).
Define a homology class C(O/K)E Ho(K) by letting C(O/K) be represented
by the simplicial 0-chain with coefficient (1/2n)Tv at each vertex vEK. Let
C ( O / K ) E HE(K, R) be the Poincar6 dual of C(O/K) (in dual cell cohomology).
It is also possible to calculate C(O/K) directly from a 2-LGF f~/K obtained
from O/K. Whereas it is intuitively more simple to calculate C(O/K) as above,
it is possible to give a more explicit formula using f~/K; such a formula, and
~/K in general, would be much easier to use on a computer than O/K.
First, suppose (a, b, c) is a triangle contained in an open hemisphere in S 2,
with sides of length a, fl and 7, and angles A, B and C. Let Na, Nb and Arcbe the
oriented normal planes to a, b and c, and let Ha,b, I-Ia,c and lib, c be the
orthogonal projections from Na to Nb, etc. Let ri:Ni - - R 2 ' b e an arbitrary
orientation preserving orthogonal map for i -- a, b and c, and let
~-~a,b = rb ~
~
-1: R2--* R2,
and similarly for fla,~ and flo,~. We want to compute Area(a, b, c) using only
the maps fl~,b, fla,c and f~b,~.
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ETHAN D. BLOCH
Isr. J. Math.
Finding the unsigned area is straightforward. First, it is not hard to see that
cos a = det(Db.c), cos fl = det(Ila,c) and cos y = det(f~a.b). Using the spherical
law of cosines for sides, it is then seen that
COSA =
cos a -- cos fl cos y
sin fl sin y
c o s a - c o s fl c o s ~,
, / 1 - c o s 2 p - c o s 2 ~, + c o s 2 p c o g ~,
det(~b,~) - det(fla,c)det(~'l.,b)
~/1 - det(~a,c) 2 - det(flb,c) 2 + det(fl..c)2det(~a,b) ~ '
and similarly for the other angles of the triangle. The unsigned area of (a, b, c )
is A + B + C - n, which is thus computable in terms of f~a,b, ~a,c and Ilb, c.
The orientation of (a, b, c ) is calculated as follows. Let (El, E2 } be an oriented
basis for R 2, and let P = (f~a,c) -1 ~
~
Then (a, b, c) is positively
oriented, negatively oriented or has zero area iffP(E~). E2 is positive, negative
or zero respectively. To see this, note that
P = ([la,c) - I ~ ~'Ib,c ~ l-la,b = ra ~ (l-la,c) -I ~ llb,c o I-!~,b o (ra)-I
: R2--)" R 2.
Therefore
e ( E , ) . E~ = (no,~) -1 o n~,c o n o , ~ ( ( r D - ' ( E l ) ) . ( r D - ' ( E ~ ) ,
since ra is an orientation preserving orthogonal map. It is now routine to check
that (Ha,c)-1o Hb,c ~ ria,b((ra)- 1(El))" (ra)- I(E2) is positive, negative or zero iff
(a, b, c) has positive orientation, negative orientation or zero area respectively. Using these calculations, it is seen that there is an explicit formula for Tv
at each vertex v ~ K , using only the collection of maps in a 2-LGF f~/K
obtained from O/K. It should be remarked that this formula for Tv in terms of
f~/K can be applied to any 2-LGF, not just one obtained from a (2,3)-LCM;
however, examples show that ifa smooth 2-plane bundle is classified in G2(R4),
then the quantities T~ computed by this formula (applied to an induced
2-LGF) will not always give the Euler class of the original bundle.
Let K be an orientable, locally embedded surface in R 3 (in particular, K must
be simplexwise embedded). Further, suppose K is suitably restricted, so that it
has a tangent (2,3)-LCM OT/K. If v ~ K i s a vertex, it is not hard to show that if
one computes T~ directly from OT/K, then T~ = A (v, K). (The issue here is that
in computing T~, one takes into account the normals to the "tangent planes" at
VOI. 67, 1989
A COMBINATORIAL CHERN-WEIL THEOREM
203
the 1-simplices of K, which are ignored in the computation of A(v, K).
Whereas these normals do affect To in the case of arbitrary (2,3)-LCM's, they
do not do so when the (2,3)-LCM is a tangent (2,3)-LCM.) If one further
assumes that wrap(v, K ) = 1, which is the case for fine enough C~-triangulations of smooth surfaces in R 3, then To = eo, which in turn equals the angle
defect at v by Theorem 2.3. Hence T oreally does generalize the standard notion
of curvature for polyhedral surfaces.
DEFINITION. Let Ol/Kand O2/Kbe simplex-normal (2,3)-LCM's. O~/Kand
02/K are hemispherically related if the same hemispheres H~, for all a E/62),
can be used in the definition of simplex-normality for both O~/K and 02/K.
Let K be a simplicial 2-complex. Suppose O~/K and 02/K are
hemispherically related, simplex-normal (2,3)-LCM's. Then C(OI/K)=
C(O2/K).
LEMMA 4.1.
PROOF. By Poincar6 duality, it will suffice to show that C(OI/K)=
C(02/K) in Ho(K, R). For each vertex v~K, let Tl~ and T2v denote the
curvatures at v of O / K and 02/K respectively. Let ,/1 and J2 denote the
simplicial 0-chains on K with coefficients (1/2re)T~o and (1/2re)T2~, respectively, at each vertex v~K. We thus need to show that ./1 and ,12 are
homologous. Using the compactness of K, we can find a sequence of simplexnormal (2,3)-LCM's Oi/K = ~ / K = r
.....
~ , / K = OJK, so that any
two consecutive elements of the sequence are hemispherically related, and
differ on at most one vertex of K'. Hence it will suffice to prove that Jt and J2
are homologous if we assume that Oi/K and 02/K differ on at most one vertex
tr* of K'. There are three possibilities, depending on the dimension of a.
First, suppose tr is a vertex of K. Then clearly T~v = T2~for all vertices vEK
other than a. On the other hand, the definition of curvature at a, together with
Theorem 2.1, imply that Tto and T2~ are independent of the values of O~/(tr)
and O2(a), and hence are equal by the assumption on O / K and 02/K.
Next, suppose a --- (v, w) is a 1-simplex of K. Then clearly T~z = T2z for all
vertices z E K other than v and w. Hence
J,
= (1/2n)(T,o-
T 3(v) + ( l / 2 n ) ( T , .
-
Let f ~ and f2~ be the maps used to compute Tj~=A(f~v, O(v)*) and T2v =
A (f2v, O(v)*), and similarly f o r f wandf2w. The pairs of mapsf~ andf2~, a n d f w
and f2w, each differ only at tr*. Let r/ and z be the two 2-simplices of K
containing a. Note that fv(r/*) = f2v(q*),fv(z*) = f2~(z*), and similarly for w.
204
ETHAN D. BLOCH
Isr. J. Math.
It is now seen that both T~ - T2~ and Ttw - T2w are equal in absolute value to
the difference of the signed areas of the geodesic triangles
(fv(q*), fl~(a*), fv(r*))
and
( f , ( q * ) , f2o(a*), f~(r*))
(this uses the fact that everything takes place in a hemisphere). Therefore, by
taking orientations into account, it is seen that Jl - .I2 = aS, where S is the
simplicial l-chain on Kwhich has coefficient (I/21t)(Tl~ - T2~) at e, and is zero
elsewhere. The proof when a is a 2-simplex is similar, where in this case S has
support only on @a.
[]
REMARK. The above lemma shows that for t w o (2,3)-LCM's to be "different," there needs to be some difference that is not hemispherical; this seems
to correspond to the fact that smooth bundles are constructed by clutching
maps, and different bundles need non-homotopic clutching maps.
5. Combinatorial Chern-Weil theorem for 2-plane bundles with even Euler
characteristic
The same assumptions about bundles used in w will apply here. Throughout
this section, let ~ be a 2-plane bundle with X(~) even. (This restriction is
because of Lemma 3.1.) We now apply the previous section to give a
combinatorial expression for the curvature and Euler class of ~.
DEFINITION. Let g:B(~)----G2(R 3) be a classifying map for ~, and let
g* : B ( ~ ) ~ S 2 be the associated map of oriented normal vectors, i.e. g ' ( x ) =
[g(x)]*. A C~-triangulation t : K - - , B ( ~ ) is called g-normal if for each 2simplex a = (u, v, w ) E K , g~(star(v,K) U star(v,K) U star(w,K)) is contained in an open hemisphere.
REMARKS. (1) For a given classifying map g, not every Coo-triangulation
K ~ B(~) is g-normal; however, a fine enough triangulation will be.
(2) If t : K ~ B(~) is g-normal, then O(~, g ) / K is simplex-normal; also, i f L
is any subdivision of K, then t: L ~ B(~) will be g-normal.
DEFINITION. Let ~ and g be as above, and let t : K - - , B ( ~ ) be a g-normal
C~176
Define the cohomology class C ( ~ ) E H 2 ( B ( ~ ) , R ) to be
C(~) = (t*) - '(C(O(~, g)/K)).
As defined, the class C ( ~ ) E H 2 ( B ( ~ ) , R) depends on (1) the choice of
triangulation t : K - - . B ( ~ ) , and (2) the choice of classifying map for ~. The
following lemma shows that C(~) in fact only depends on the bundle ~ itself.
Vo1. 67, 1989
A COMBINATORIAL
CHERN-WEIL
205
THEOREM
L~MMA 5.1. Let ~ be a 2-plane bundle over a surface with Z(~) even. Let
g : B(~) --- G2(R3) be any classifying map for ~, and let t : K ~ M be a g-normal
C~-triangulation .
(i) The cohomology class (t*)-I(C(O( ~, g)/K))EH2(B( ~), R) is invariant
under subdivision of K.
(ii) (t*)- ~(C(O(~, g)/K)) depends only on the bundle ~.
PROOF. (i) Let L be a subdivision of K. Order the vertices of K, and let
sd : {0-chains of K} ---, {0-chains of L} be as described in [F, p. 14]. The map
sd. : Ho(K)~ Ho(L) is an isomorphism; to prove this part of the lemma it will
suffice to show that sd.(C(O(~, g)/K))--C(O(~, g)/L). If C~ denotes the
curvature at vertex v E K with respect to O(~,g)/K, then C(O(~,g)/K) is
represented by a 0-chain in K that is (1/2n)T v at v. sd.(C(O(~, g)/K)) is
represented by the 0-chain in L that is (1/2n)Tv at vertices v E L which are
vertices in K, and 0 at other vertices o f L . Let ~/L be the (2,3)-LCM defined as
follows. If r ~ L is a simplex, and ~/~ K is the minimal dimension simplex of K
containing r, then let @(z*) = O(~/*), where O here is O(~, g)/K. It is easy to
check that: (1) @/L and O(~, g)/L are hemispherically related, and (2) C(@/L)
is represented by the same 0-chain as sd.(C(O(~, g)/K)). The desired result
now follows from Lemma 4.1.
(ii) First, it follows immediately from (i) that (t*)-~(C(0(~, g)/K)) is independent of the choice of g-normal C~-triangulation, since any two C ~triangulations of B(~) have a common C~-subdivision, which must be gnormal (see [W]).
Next, suppose g, h : B ( ~ ) - " G2(R3) are classifying map for ~. Since we may
subdivide our C~-triangulations, it is possible to choose a C~
t : K -- B(~) which is both g- normal and h- normal. By Lemma 3.1, g and h are
smoothly homotopic. Let F,:K X [0, 1]---B(~) be such a homotopy, i.e.
F0 = g and F~ = h. Subdividing K further if necessary, we may assume that for
all t ~ [ 0 , 1], t : K ~ B ( ~ ) is F,-normal. By the compactness of K, there are
numbers 0 = t~ < t2 ~ " " " ~ t, = 1 such that O(~, Ft,)/K and O(~, Ft,+,)/K are
hemispherically related for i = 1, 2 , . . . , n - 1. Lemma 4.1 now implies that
C(O(~, g)/K) = C(O(~, Ft,)/K) -- C(O(~, Ft)/g) . . . . .
and the lemma is complete.
C(O(~, h)/K),
[]
LEMMA 5.2. Let ~ be a 2-plane bundle over a surface with Z(~) even. Then
(i) if ~ has a non-zero section, then C(~) = 0;
(ii) if h : N---, B( ~) is a smooth map, then C(h*( ~)) = h*(C( ~)).
206
ETHAN D. BLOCH
Isr. J. Math.
PROOF. (i) Suppose ~ has a non-zero section. This implies that ~ is the
Whitney sum of a trivial line bundle and some other bundle. Since ~ is an
oriented 2-plane bundle, this other sub-bundle must also be an oriented line
bundle, i.e. it is trivial. Hence ~ is trivial. Since C(~) can be computed with any
classifying map, by Lemma 5.1(ii), use a constant map g:B(~)-,G2(R3).
Clearly Tv = 0 for all vertices vEK, and hence C(~) = 0.
(ii) Let g: B(~)--G2(R 3) be a classifying map for ~, and give h*(~) the
induced classifying map g oh. Let t : K - , B ( ~ ) be a g-normal C~
lation, and let s : L - - , N be a (g o h)-normal C~176
Let Lt be a
subdivision of L such that h has a simplicial approximation ht with respect to
triangulations L~ and K(as in [S] p. 126). Note s : LI ---Nis (g o h)-normal, as
previously remarked; i f K and L are chosen fine enough, then we may assume
that s: L~--, N is also (g o hl)-normal. Since hl*(~) ~ h*(~), it suffices to prove
the lemma for h~. This follows from the definition of h* on the (dual cell)
cochain level. Let V~Ll be a vertex, and let D(v) be the 2-cell dual to v. If
hi(v) = w, for some vertex w ~ K , then hl(D(v))C D(w), and h~(OD(v))c
OD(w). For each v ~ L I , let degv be the degree of the map h I [ D ( v ) " D(v)--"
D(w). Let J denote the dual cell 2-chain on Kwith value (1/2n)T~ on D(v), for
each vertex v~K, i.e. J represents C(~). Then h*(C(~)) is represented by
h~(J), which is given by
hl#( J)(O(v)) = (deg~)J(D(n~(v))),
for v~ Lt.
However, it is not hard to see that for vE L t, the definition of T~ with respect to
h*(~) implies that T~ = (deg~)Th,~v), and the lemma now follows.
[]
THEOREM 5.3.
I f ~ is a 2-plane bundle over a surface with Z.( ~) even, then
= z(O.
PROOF. By combining Theorem I. 11-30 with the proofs of Corollaries
V.13-24 and V.13-25, all in [Sp], we obtain the following fact: if to every
smooth, oriented m-plane bundle ~ over a smooth, oriented, closed manifold,
m even, there is associated a cohomology class E ( ~ ) ~ H m(B(~), R) such that
E ( O commutes with pull-backs, and such that E(~) - 0 whenever ~ has a
non-zero section, then E(~) = AmZ(~) for some constant Am that only depends
on the dimension m. The proof works for each dimension separately. Therefore, using Lemma 5.2, C ( ~ ) = A 2 ) ( ~ ) for some constant A2 that does not
depend on ~.
It remains to show that A2 = 1; computing one example will suffice. Let
TS2-,.S 2 be the tangent bundle of the unit 2-sphere in R 3. Note that
Vol. 67, 1989
A COMBINATORIAL CHERN-WEIL THEOREM
207
C(TS2)([S2]) = 2. TS 2 is classified by the identity map i : S 2 - ' ~ S 2 ~ G2(R3). Let
Kbe a regular tetrahedron centered at the origin in R 3, and with vertices on S 2.
Then radial projection t: K - , S 2 is an /-normal C~176
By convexity, it is easy to see that
Y~ Tv = 4n
( = the area of $2),
vEK
and hence
C(TS~)([S2]) =
Y~ ( 1 / 2 n ) T ~ = 2.
vEK
S i n c e z(TS2)([S2])
6.
= x ( S 2) =
2, A 2 = 1.
[]
Area for spherical cones - - proof of Theorem 2.1
If C is a convex geodesic polygon in a hemisphere o f S 2, with angles ~k at its
vertices (k = 1. . . . . n), then the area bounded by the polygon is known to be
kzln ~k - - (n - 2)n. The main step in the proof of Theorem 2.1 is a generalization of this formula to the case where C is not necessarily convex or immersed.
We will use the notation of w
DEFINITION. Let C n = (a, . . . . , a,) be a triangulation of S ~, with a given
orientation, and let f : 6",---S + be an SG map (where S + is an open hemisphere in $2). Furthermore, assume f is injective on each 1-simplex of C~.
Define S: c S + to be the set
S: = S + -
G
(great circle containing f((ak, ak +0)),
kzl
where addition is mod(n). Let kbe in { 1 , . . . , n }. To each x E S / w e associate a
quantity t k ( f , X) as follows. The geodesics containing (f(ak-~), f(ak)) and
(f(ak),f(ak+O) divide S + into four regions (two of which might be degenerate). We label the regions I, II, III, IV as in Fig. 6.1; the two cases in the
figure correspond to whether ( f(ak + l), f(ag _ i), f ( a k ) ) has positive or negative
orientation.
Elk(f, X) is now defined as follows, where the orientation used is that of
( f(ak+ 1), f(ak-~), f(a~)):
208
ETHAN
f(ak-I)
D. B L O C H
f(ak+,)
Isr. J. M a t h .
-
II
-"
9
I
".
f(ak)
. .. "
o
III
., . .
,,,,
9
9 " f(ak) 9
.
9
II
~
f(ak-O
positive orientation
(
a
k
+1)
negative orientation
Fig.
/~k(f, x ) =
f
6.1.
a(flak +1), flag), f(ak-,)) -- n
x E I U II, pos. orientation,
a(f(a, +3, f(ak), f(ak-l)) + 7t
x E I U II, neg. orientation,
La(f(ak +l), f(ak), f(ak-3)
x E III U IV, any orientation.
Finally, for x ESf, let w(f, x) denote the winding number of f about x.
THEOREM 6.1. Let f: C,---,S + be an SG map (with notation as above),
Assume f is injective on each 1-simplex in C. ; let x ~ SI. Then
A ( f , x) = ~ ilk(f, X) + 2rtw(f, X).
k=l
PROOF. For each k E{1 . . . . , n} let Sk = + 1 depending on whether
has positive or negative orientation. Suppose
(x, f(ak), f(ak+ 1)) has (postive) angles 5k at f(ak), ek at f(ak+ ~), and ~/k at x.
Then
Area(x, f( ak ), f( ak +l) ) = Sk { Sk + ek + ?Ik -- it)
(X, f(ak), f(ak+ I ) )
= sk ((5, - ( ~ / 2 ) ) + (e~ - ( ~ / 2 ) ) + ~k}.
It is easy to see that Z~_ 1 Skl~k = 2nw(f, x). Next, consider the verticesf(ak); it
is claimed that fig(f, x) = Sk(Sk -- (n/2)) + Sk-l(ek-I -- (~/2)). Assuming the
claim is true, the theorem follows because
A ( f , x) = ~ Area(x, f(ak), f(ak+O)
k-l
=
~: s ~ { ( ~ - ~ / 2 ) + (e, - ~ / 2 ) +
k-I
~,}
Vol. 67, 1989
209
A COMBINATORIAL CHERN-WEIL THEOREM
n
_n
= Y~ {Sk(Ok -- ~Z/2) + S k - , ( e k - , -
~/2)} + ~ Skrlk
k=l
k=l
= X M f , x) + 2 w(f, x).
k=l
It remains to prove the claim made above. Let k E { 1. . . . , n } be fixed.
There are four cases, depending on whether each of S k - 1 and Sk are 1 or -- 1.
Case 1. S k - 1, Sk = 1. There are two possibilities: either x is in region I and
(f(ak+O,f(ak-3,f(ak))
is positively oriented, or x is in region II and
( f ( a k + ~), f(ak-~), f ( a k ) ) is negatively oriented. In the former case, it is seen
that a ( f ( a k + i), f(ak), f(ak-1)) = Ok + ek-I (see Fig. 6.2(i)). Hence
Sk(0k -- Zt/2) + S k - l ( e k - i -- Zt/2) = Ok +
ek-I
--
~'~
= a ( f ( a k + l), f(ak), f ( a k - I ) ) -- rt
= flk(f,x),
by the definition of f l k ( f , x ) in this case. If x is in region II and
( f(ak + O, f ( a k - O , f l a g ) ) is negatively oriented, then
a ( f ( a k + l), f(ak), f ( a k - i ) ) = -- (22 -- (Ok + ek-l)) = Ok + ek-, -- 2rt,
the negative sign coming from the fact that we are using signed angles (see Fig.
6.200). Hence
Sk(Ok -- rt/2) + Sk_l(~,k_l
-
-
rr/2) = Ok + ek-, -- n = (Ok + ek-, -- 2rt) + ~r
= a ( f ( a k + i), f(ak), f(ak - 3) + rr = ilk(f, X),
by the definition o f ilk(f, X) is this case.
x
x
f(al,-,)
1
f(~+O
L
.. "J[.ak)"" "-,
f ( a , _ y
~'9~f S s p"
Jta, J
t
J
I
(i)
(ii)
Fig. 6.2.
~.~ak+l)
210
ETHAN D. BLOCH
Isr. J. Math.
Case 2. Sk-1, Sk = -- 1. This case is similar to the previous case.
Case 3. S k - l = l , S k = - - l . Point x must be in region IV, though
(f(ak+ O, f(ak-1), f(ak)) could be either positively or negatively oriented. In
the former case, it is seen that a(f(ak+O, f(ak), f(ak-l))= e k - i - ~r (see Fig.
6.3(i)). If ( f(ak+ O, f(ak-~), f(ak)) is negatively oriented, then it still holds that
a( f(ak +O, f(ak), f(ak-1)) = ek-~ -- ~k (see Fig. 6.3(ii)). It is now seen that
Sk(ak -- (n/2)) + Sk-l(ek-t -- (~/2)) = ilk(f, X)
similarly to the previous cases.
X
f( ak+l)
X
~
f( ak- 1 ) ~ l
--_
f(ak)
Ok t _ > ' - . .
J f ( a k )
"-
--
f(at-,)
f(ak +,)
(i)
(ii)
Fig. 6.3.
Case 4. Sk-1 = -- 1, Sk = 1. This case is similar to the previous case.
[]
We note that the formula in Theorem 6.1 reduces to the standard formula
Y-17=, 7k -- (n -- 2)n whenever f ( C , ) is a convex, embedded curve. I f f ( C , ) is a
convex, embedded curve, then it bounds a well defined region in S+; choose
any point x in this region. It is easy to see that (1) w(f, x) = 1, and (2) for each
k E { 1 . . . . . n}, x is in region I, and (f(ak+l),f(ak-O,f(ak))
orientation. Hence
has positive
ilk(f, X) = a( f(ak +,), f(ak), f(ak- ,)) -- n = ~'k -- n.
It now follows that
n
t k ( L X) "~ 2 n w ( f , X) = ~ (~'k -- rt) + 2n = Y~ ?k -- (n -- 2)rt
k-I
k~l
k=l
as desired.
PROOF OF TrIEORV.M 2.1.
We m a y assume W L O G that f is injective on
VOI. 67, 1989
A COMBINATORIAL CHERN-WEIL THEOREM
211
each interval of Cn. ( I f f is not injective on some interval, then it maps that
interval to a point, and hence the cone from x to the image of this interval has
zero area in S+; thus f : Cn--'S + can be replaced by the obvious map
f ' : Cp ~ S+, for some p < n.) Syis the disjoint union of (geodesically convex)
components. It is easy to see that the functions ilk(f, X) and w ( f , x ) are
constant on each of the components of Sy (although they are not necessarily
constant on all of Si). Theorem 6.1 then implies that A ( f , x ) is constant on
each of these components. However, A (f, x) is defined and continuous on all
o f S § (this uses the fact that everything takes place in an open hemisphere). It
now follows easily that A (f, x) is constant for all x in S §
[]
7. Proof of Theorem 2.3
Before giving the proof of Theorem 2.3, we need the following preliminaries.
As in w assume Kis a star normal simplexwise embedded surface in R 3, and G
is a combinatorial Gauss map on K. For each vertex v ~ K , let f : Cp ---S6(v) be
as in the definition of eo (in w
DEFINITION. Let v E K be a vertex. Label the 2-simplices in star(v, K) by
r/l, r h , . . . , r/p, in the order corresponding to the orientation of K. Assume v is
at the origin of R 3, and extend the r/k until they intersect S 2. Call the arcs of
intersection gk. We say gk and gk +z intersect positively (respectively negatively)
if the angle from gk+l tO gk centered at their intersection is positively (resp.
negatively) oriented in S 2. For each k ~ { 1 , . . . , p }, we say v is o f type I, 2, 3 or
4 at k as follows:
Type
Type
Type
Type
1:
2:
3:
4:
gk- 1 and
gk- J and
gk - ~ and
gk- ~ and
gk intersect negatively; gk and gk +~ intersect negatively.
gk intersect positively; gk and gk + ~ intersect positively.
gk intersect negatively; gk and gk + ~ intersect positively.
gk intersect positively; gk and gk +~ intersect negatively.
See Fig. 7.1. As in w let Sx c S 2 denote the open hemisphere centered at x.
Now, fixing k, recall that the geodesics containing <G(r/~'_l), G(r/g)) and
(G(r/~'), G(r/~+ ~)) divide S~v~ into four regions (two of which might be degenerate), called regions I, II, III, IV. We say v is well-matched at k if v is either of
type 1 or 2 at k, and G(v) is in regions I or II, or if v is of type 3 or 4 at k, and
G(v) is in regions III or IV; otherwise we say v is missmatched at k. Finally, we
define the local error term ek to be
212
ETHAN D. BLOCH
07t
ek =
Isr. J. Math.
if v is well-matched at k,
if v is missmatched at k.
The global error term is
1 _p
e(v, G) = ~ Y. ek ----89(number of missmatches in star(v, K)}.
k-l
type 1
type 2
type 3
type 4
Fig. 7.1.
LEMMA 7.1. Let K be a star normal simplexwise embedded surface in R 3,
and let G be a combinatorial Gauss map. Let v E K be a vertex, let f : Cp ~ S~t~)
be as above. Choose k ~ { l , . . . , p }, and suppose qk has angle Otk at v.
(i) I f v is o f type 1 or 2 at k, then (1) (f(q~+O,f(q~_l),f(rl~)) is positively
oriented, (2) a point x E S 2 is in regions I or H iff the plane through the origin
orthogonal to x does not interesect gk, and (3) a(f(r/~'+l), f(r/~'), f(r/~'_x))=
7C - -
t2 k.
(ii) I f v is o f type 3 or 4 at k, then (1) ( f(q~+ ~), f(q~_ ~), f(q~) ) is negatively
oriented, (2) a point x ~ S 2 is in regions III or I V iff the plane through the origin
orthogonal to x does not intersect gk, and (3) a(f(tl~+,), f(q~), f(q~_ ~)) = - ak.
(iii) Define ilk(f, G(V)) as in w Then ilk(f, G(v)) = - ak + ek (no matter
what type v is at k ).
PROOF. We will consider the case where v is of type 1 at k; all other cases
are similar. The lemma is trivial if everything is written out in spherical coordinates (p, 0, O). WLOG, assume that v is at the origin, and that gk is in the
x - y plane, and has endpoints a = (1, - ak/2, ~12) and b = (1, ak/2, n/2).
Suppose that the angle between gk- ~and gk is 2, and the angle between gk and
gk+~ is/t. It is seen that f(r/~') = G(q*) = (I, 0, 0),
f(tl~-z) --- G(q~'_,) = (1, - (ak + 7t)/2, 2),
and
f(tl~-i) = G(q~_~)= ( 1 , - (ak + lt)/2, 2),
the results in (i) now follow easily.
Vol. 67, 1989
A COMBINATORIAL CHERN-WEIL THEOREM
213
To show (iii) in case v is of type 1 at k, note that the definition of
fig(f, G(v)) depends on which region G(v) is located in with respect to the
geodesics containing (f(q~'_ 1), f(q~')) and (f(t/*), f(q~'+ 1)7, and on the orientation of (f(q~'+ 1), f(q~'-1), f(q~)). We have just seen that this orientation is
positive, and that
a( f( tl~+,), f( tl* ), f( tl~-1)) = n - ak.
If G(v) is in region I or II, then ek = 0 (since v is well-matched in this case), and
thus by definition
fig(f, G(v)) = a(f(q~+ ,), f(tl~), f(q?_ ,)) - rc = (n - ak) - rc
_-- - - a k + 0 - - _ . - - a g + e k.
If G(v) is in region III or IV, then eg = n (since v is missmatched in this case),
and thus by definition
fig(f, G ( v ) ) = a ( f ( t l ~ + l ) , f ( t l ~ ) , f ( t l ~ - i ) ) = l r
-at
= - - a k + ek.
[]
LEMMA 7.2. Let K be a star normal simplexwise embedded surface in R ~,
and let G be a combinatorial Gauss map. Let v ~ K be a vertex, and let
f: Cp o So(v) be as above. Then wrap(v, K) = w(f, G(v)) + e(v, G).
PROOF. Assume WLOG that v is the origin of R 2, and G(v) is the North
pole of S 2. Consider the geodesic polygon C = gl t3 g2 (J 9 9 9 U gp. Label the
vertices of this polygon (a~, a 2 , . . . , ap }, so that gk = (ag, ag +1). Note that the
map f : Cp ~ Sa~) is determined by C and the point G(v), and hence so are
wrap(v, K), w(f, G(v)) and e(v, G). Now, we deform C by pushing its vertices
"down," staying on or near their original longitudes, so that they all end up in
the open Southern Hemisphere, and during the process the gk maintain their
orientations with respect to the North Pole. We can chose this homotopy to be
the composition of a sequence of smaller homotopies, such that in each of
these smaller homotopies only one vertex ak moves, and one of the following
three cases occurs: (1) a single arc(f(q~_ 1), f(t/~')) passes through G(v), but all
pairs gg and gg +1 maintain the positivity or negativity of their intersection, (2)
no arc(f(tl~_O,f(q?,')) passes through G(v), but the pair gk and gk+l which
intersect in the moving vertex change the positivity or negativity of their
intersection, though all other pairs gk and gg+~ maintain the positivity or
negativity of their intersection, or (3) no arc(f(t/~_ ~), f(q~')) passes through
G(v), and all pairs gg and gg+l maintain the positivity or negativity of their
intersection. Such a homotopy will not change wrap(v, K). w(f, G(v)) and
e(v, G) certainly may change during the homotopy. We will indicate in the next
214
ETHAN D. BLOCH
Isr. J. Math.
paragraph that whereas w(f, G(v)) and e(v, G) may each change, the sum
w(f, G(v)) + e(v, G) remains constant. To conclude the proof, it remains to
note that if C is entirely in the open Southern Hemisphere, then wrap(v, K) =
w(f, G(v)), and e(v, G ) = 0. The former claim is straightforward, and the
latter follows from part (2) in Lemma 7.1 (i) and (ii), which show that ek -- 0
for all k.
We need to show that the sum w(f, G(v)) + e(v, G) remains constant during
any of the three types of smaller homotopies described in the previous
paragraph. Clearly neither w(f, G(v)) nor e(v, G) changes during a type (3)
homotopy. In a type (2) homotopy, it is evident that w(f, G(v)) is constant. We
omit the details here, but it is not hard to show that all the ekremain constant as
well; the point is that for both arcs gk and gk+~ containing the moving vertex,
the type of v changes from 1 or 2 to 3 or 4 (or vice versa), and the corresponding
region in which g(v) is located changes from I or II to III or IV (or vice versa),
thus not changing whether v is well-matched or missmatched at k and k + 1.
Finally, for type (1) homotopies, we will show that when a geodesic containing
an arc (f(r/~'_l),f(q~')) passes through G(v), the sum w(f,G(v))+e(v,G)
remains constant. Call this geodesic t. By budging the homotopy slightly, we
may assume that when G(v) crosses t, it does not do so at f(q~'_~) or f(r/*), v
could be of any type at k - 1 and k; there are four cases, depending on whether
v is of type 1 or 2, or of type 3 or 4, at each o f k - 1 and k. We will examine the
case where v is of type 1 or 2 at k - 1, and is of type 3 or 4 at k; the other three
cases are similar. It is easier to think of G(v) crossing t, rather than vice versa.
There are generically three types of crossings, labeled as a, b and c in Fig. 7.2.
In situations a and c, G(v) does not cross the image o f f , so that w(f, G(v)) is
unchanged. In situation a, G(v) moves from region III to region II at k - 1 (so
ek-~ decreases by n, being of type 1 or 2), and it moves from region IV to region
I at k (so ek increases by n, being of type 3 or 4). Thus
1
P
remains unchanged in situation a. Similarly in situation c. Hence, in both
situations a and c, w(f, G(v)) + e(v, G) remains unchanged. In situation b,
G(v) crosses (f(q~'_ i), f(q~')) so that (G(v), f(q~_ ~), f(q~')) changes from being
positively oriented to being negatively oriented (the other direction of crossing
is similar). Hence w(f, G(v)) decreases by 2ft. However, G(v) moves from
region I to region IV at k - 1 (so ek-l increases by n, being of type 1 or 2), and
Vo1. 67, 1989
215
A COMBINATORIAL CHERN-WEIL THEOREM
it moves from region IV to region I at k (so ek increases by 7t, being of type 3 or
4). Thus e(v, G) increases by 2re in situation b, and hence w(f, G(v)) + e(v, G)
remains unchanged.
[]
II
a
b
1
f'(q~-O
II
" "-
IV
t
IV
~" - ,..
f'(r/~)
".
c
.....
l-
I
Fig.
7.2.
PROOF OF THEOREM 2.3. Let {~/1. . . . . qp) be the 2-simplices of star(v, K)
in order corresponding to the orientation of K. To simplify our situation, we
may assume t h a t f is injective on each interval of Cv. Suppose conversely that
f(t/*) =f(t/~'+l) for some k. Then the adjacent 2-simplices t/k and t/k+1 are
parallel. We could then modify star(v, K) by replacing qk and/~k + 1 by a single
2-simplex in the obvious way (such a 2-simplex might not "fit" in K anymore,
but the curvature ev only depends on star(v, K)). Continuing in this way, we can
eliminate all cases where f is not injective on an interval. It is not hard to see
that this process would change neither A (v, K) nor wrap(v, K). Hence we may
assume WLOG that f i s already injective on each interval.
L e t o/k be the angle at v in 2-simplex ~k. For each k ~ { 1. . . . . p }, we can
define ilk(f, G(v)) as in w Then
ev=A(v, K) -- 2n[wrap(v, K) -- 1]
= { k=,
~ ilk(f, G(V))+ 2nw(f, G ( v ) ) } - 2n{w(f, G(v))+e(v, G ) - 1 )
by Theorem 6.1 and Lemma 7.2
= Y, ( - ak + ek) -- 2re(V, G) + 2~ by Lemma 7.1(iii)
k~l
=t
o,t
k=l
216
ETHAN D. BLOCH
where the last line follows from the previous one because Zg_ l
by the definition ofe(v, G).
Isr. J. Math.
~k =
2he(v, G),
[]
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